#72277152

order	13688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₉ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅₉ ⋊ C₂×C₂₉
Hall	C₅₉⋊C₂₉ ⋊ D₄
Hall	C₂²⋊D₅₉ ⋊ C₂₉
Sylow	C₅₉ ⋊ D₄×C₂₉

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×AGL(1,𝔽₅₉)
commutator, socle, cosocle	C₂×C₅₉	C₂²×C₂₉
Fitting	C₂²×C₅₉	C₂×C₂₉
hypercenter	C₂²	AGL(1,𝔽₅₉)
hypocenter	C₅₉	D₄×C₂₉
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 236

Polycyclic presentation:

2^2,29,2,59|b^a=bd,e^b=e^58,e^c=e^28

Conjugacy classes & character table

149 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×AGL(1,𝔽₅₉)
maximal normal subgroup of	D₄×AGL(1,𝔽₅₉)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 29, 2, 59]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^58);;
gap> SetConjugate(coll, 5, 3, F.5^28);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);